A group of 12 men can do a piece of work in 14 days and other group of 12 women can do the same work in 21 days. They begin together but 3 days before the completion of work, man's group leaves off. The total number of days to complete the work is:

Options

A. `65//4`
B. `93//3`
C. `51//5`
D. 60

Answer

To solve the problem step by step, we can follow these calculations:

### Step 1: Determine the work done by men and women in one day.
- **12 men can complete the work in 14 days.**
- Work done by 12 men in one day = \( \frac{1}{14} \)
- Work done by 1 man in one day = \( \frac{1}{14 \times 12} = \frac{1}{168} \)

- **12 women can complete the work in 21 days.**
- Work done by 12 women in one day = \( \frac{1}{21} \)
- Work done by 1 woman in one day = \( \frac{1}{21 \times 12} = \frac{1}{252} \)

### Step 2: Calculate the total work done by men and women together in one day.
- Work done by 12 men and 12 women together in one day:
- Work done by 12 men = \( \frac{1}{14} \)
- Work done by 12 women = \( \frac{1}{21} \)

To combine these, we find a common denominator (which is 42):
- Work done by 12 men in one day = \( \frac{3}{42} \)
- Work done by 12 women in one day = \( \frac{2}{42} \)

Total work done in one day by both groups:
\[ \text{Total work in one day} = \frac{3}{42} + \frac{2}{42} = \frac{5}{42} \]

### Step 3: Set up the equation for total work done.
Let \( X \) be the total number of days to complete the work. Since the men leave 3 days before the work is completed, they work for \( X - 3 \) days.

- Work done by men in \( X - 3 \) days:
\[ \text{Work by men} = \frac{5}{42} \times (X - 3) \]

- Work done by women in \( X \) days:
\[ \text{Work by women} = \frac{2}{42} \times X \]

### Step 4: Combine the work equations.
The total work done by both groups must equal 1 (the whole work):
\[ \frac{5}{42}(X - 3) + \frac{2}{42}X = 1 \]

### Step 5: Solve the equation.
Multiply through by 42 to eliminate the fraction:
\[ 5(X - 3) + 2X = 42 \]
Expanding gives:
\[ 5X - 15 + 2X = 42 \]
Combine like terms:
\[ 7X - 15 = 42 \]
Add 15 to both sides:
\[ 7X = 57 \]
Divide by 7:
\[ X = \frac{57}{7} = 8.14 \text{ days} \]

### Step 6: Final answer.
The total number of days to complete the work is \( \frac{57}{7} \) days, which is approximately 8.14 days.

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